Quantum Hall effect at half-filled Landau level: Pairing of composite fermions

We discuss the possibility of the quantum Hall effect at half-filled Landau level in terms of the pairing of the composite fermions. In the absence of Coulomb energy, we show that the ground state of the system is described by the {\it p}-wave BCS pairing state of composite fermions. When the ratio $\alpha \equiv (e^2/\epsilon \ell_B)/\epsilon_F$ ($\ell_B$ : the magnetic length, $\epsilon_F$ : Fermi energy of the composite fermions) is larger than a critical value $\alpha_c \sim 8.2$ the gap of the pairing state vanishes. However, $\alpha$ remains less than $\alpha_c$ if $\hbar \omega_c \gg e^2/\epsilon \ell_B$ holds. Then in this situation it is possible that the pairing state which results in the quantum Hall effect occurs. The effect of the real spin degrees of freedom and the Zeeman energy is also discussed.